Displacement of ball bearing using conservation of energy

[phosgene]

Homework Statement

A large ball bearing is suspended as a pendulum. One end of the pendulum is held by the electromagnet (1) and the ball is initially held by magnet (2) at some height, h, above its lowest position. The ball is released from (2) and as the pendulum swings through the vertical, the ball cuts an infrared beam (3) and causes the electromagnet holding the string (1) to release it. Ideally, the string is released at the instant the ball cuts the infrared beam. The ball then falls a height L and travels a horizontal distance D from the point of release.

The final displacement (D) of the ball from the release point can be determined using:

$D^2=4hL$

I can't figure out how this was derived.

Homework Equations

Potential energy = $mgh$
Kinetic energy = $\frac{1}{2}mv^2$

The Attempt at a Solution

I really have no idea. I thought that the final displacement from the release point would be dependent only on h, as the force of gravity after release is only acting downwards, so the horizontal velocity wouldn't change. I've also tried 'reverse-engineering' the equation to figure out how it was derived, but I don't understand where the L*h value came from, as the final kinetic energy should be mg(L+h).

 

[ehild]

The velocity of the ball is horizontal at point 3 and you get it from conservation of energy. mgh=1/2 m v^2. After reaching point 3, the ball is a projectile.

ehild

 

[grzz]

Try the solution in two steps.
a) falling through h
b) falling through L.

 

[Quinzio]

I thought that the final displacement from the release point would be dependent only on h, as the force of gravity after release is only acting downwards, so the horizontal velocity wouldn't change.

Ok, horizontal velocity doesn't depend on L, but what if L is very big, let's say 1 mile ?
The ball takes "a lot of" time before touching the ground and during that time it moves horizontally as well.

 

[Shivam123]

Ok, horizontal velocity doesn't depend on L, but what if L is very big, let's say 1 mile ?
The ball takes "a lot of" time before touching the ground and during that time it moves horizontally as well.

Yes it does..So where's the problem?